6.006 Recitation. Build

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1 6.006 Recitation Build

2 Outline Asymptotic Notation Merge-Sort Python Cost Model (maybe) docdist{5,6}.py

3 Asymptotic Notation f1: 1,000 * n ,000,000 F1 F2 F3 f2: n 375,000, ,000, ,000,000 f3: 10,000 * n mm 190mm 310mm

4 Asymptotic Notation f 1.1 * g f = O(g)

5 Asymptotic Notation f 0.9 * g f = Ω(g)

6 Asymptotic Notation f 0.9 * g 1.1 * g f = Θ(g)

7 Asymptotic Drowning lg(n 100 ) = 100 * lg(n) = Θ(lg(n)) lg(n 0.1 ) = 0.1 * lg(n) = Θ(lg(n)) lg5(n) = lg(n) / lg(5) = Θ(lg(n)) n lg(5) = n so n 2 < n lg(5) < n 3

8 Asymptotic Headaches (atoms in the universe) lglog5(lg lg100 n) (20n) 7 5 lg3 n n 2 + (lg3)n lg( ) N N/2

9 Stirling Banishes the Evil N! ~= sqrt(2πn) * ((N/e) ^ N) Substitute in lg( N ) N/2 Reduce terms, obtain O(N)

10 Binary Search for

11 Divide and Conquer 1. Divide Break into smaller subproblems 2. Conquer Really small subproblems are easy 3. Profit Combine answers to sub-problems

12 Merge-Sort 1. Divide Break into 2 equal-sized sublists 2. Conquer 1-element lists are sorted 3. Profit Merge sorted sublists

13 Just Merge L1 = [2, 3, 5, 7] L = [] L2 = [2, 4, 8, 16]

14 Just Merge L1 = [3, 5, 7] L = [2] L2 = [2, 4, 8, 16]

15 Just Merge L1 = [3, 5, 7] L = [2, 2] L2 = [4, 8, 16]

16 Just Merge L1 = [5, 7] L = [2, 2, 3] L2 = [4, 8, 16]

17 Just Merge L1 = [5, 7] L = [2, 2, 3, 4] L2 = [8, 16]

18 Just Merge L1 = [7] L = [2, 2, 3, 4, 5] L2 = [8, 16]

19 Just Merge L1 = [] L = [2, 2, 3, 4, 5, 7] L2 = [8, 16]

20 Just Merge L1 = [] L = [2, 2, 3, 4, 5, 7, 8, 16] L2 = []

21 Running Time Analysis Binary Search Merge Sort Subproblems 1 2 Subproblem size N / 2 N / 2 Time to Divide Time to Profit Θ(N) T(N) T(N/2) + 2T(N/2) + Θ(N)

22 Recursion Tree Analysis Binary Search Merge Sort T(N) T(N/2) + 2T(N/2) + Θ(N) Tree depth lg(n) lg(n) Cost per level Θ(N) Total cost Θ(lg(N)) Θ(N*lg(N))

23 Python Cost Model Motivation change + to extend for 1000X speed Approach stand on the shoulders of giants (Ron) focus on the asymptotic cost

24 Timing.py a.k.a. Ron s Shoulders 1 print "Test List-11: Sort" 2 spec_string = "1000<=n<=100000" 3 growth_factor = 2 4 print "Spec_string: ",spec_string, "by factors of", growth_factor 5 var_list, param_list = make_param_list(spec_string,growth_factor) 6 # f_list = ("n","1") 7 f_list = ("n*lg(n)",) 8 run_times = [] 9 trials = for D in param_list: 11 t = timeit.timer("l.sort()", "import random;l=[random.random() for i in range(%(n)s)]"%d) 12 run_times.append(t.timeit(trials)*1e6/float(trials)) 13 fit(var_list,param_list,run_times,f_list)

25 Pset Hint Good artists borrow, great artists steal Steve Jobs, CEO Apple Inc. quoting Pablo Picasso

26 Python Lists L, M have n items Creation list() Θ( 1 ) Access L[i] Θ( 1 ) Append L.append(0) Θ( 1 ) Concatenate L + M Θ( n ) Pop L.pop() Θ( 1 ) Delete first del L[0] Θ( n )

27 Python Lists II L, M have n items P has n/2 items Slice extraction L[0:n/2] Θ( n ) Slice assignment L[0:n/2] = P Θ( n ) Copy L[:] Θ( n ) Reverse L.reverse() Θ( n ) Sort L.sort() Θ( n * lg(n) )

28 Python Strings s, t have n characters Creation list() Θ( 1 ) Extract a char s[i] Θ( 1 ) Concatenate s + t Θ( n ) Extract substring of n/2 characters s[0:n/2] Θ( n )

29 Python Dictionaries D has n items Creation dict() Θ( 1 ) Access D[i] Θ( 1 ) Copy D.copy() Θ( n ) List items D.items() Θ( n )

30 Python Cost Exercise 1 def merge(l,r): 2 i = 0 3 j = 0 4 answer = [] 5 while i<len(l) and j<len(r): 6 if L[i]<R[j]: 7 answer.append(l[i]) 8 i += 1 9 else: 10 answer.append(r[j]) 11 j += 1 12 if i<len(l): 13 answer.extend(l[i:]) 14 if j<len(r): 15 answer.extend(r[j:]) 16 return answer Θ(N) Θ(N) Θ(N) Θ(N) ))) Θ(N) ))) Θ(N) ))) ))) Θ(N) Θ(N) ))) ))) 1 1 1

31 Python Cost Exercise II 1 def merge_sort(a): 2 n = len(a) 3 if n==1: 4 return A 5 mid = n//2 6 L = merge_sort(a[:mid]) 7 R = merge_sort(a[mid:]) 8 return merge(l,r) Θ(N) Θ(N) Θ(N) T(N/2) T(N/2) 1

32 Python Arithmetic x, y, z have n bits Addition x + y Θ( n ) Subtraction x - y Θ( n ) Multiplication x * y Θ( n ) Division x / y Θ( n 2 ) Modular Exponentiation powmod(x, y, z) Θ( n 3 ) Power of 2 2 ** n Θ( 1 )

33 Python Arithmetic x, y, z have n digits Addition x + y Θ( n ) Subtraction x - y Θ( n ) Multiplication x * y Θ( n ) Division x / y Θ( n 2 ) Modular Exponentiation powmod(x, y, z) Θ( n 3 ) Power of 2 2 ** n Θ( 1 )

34 And we re done! (617) , no voic AIM: victorcostan Google Talk: 32G-8th Floor